Influence of the surrounding medium on the luminescence-based thermometric properties of single Yb3+/Er3+ codoped yttria nanocrystals

While temperature measurements with nanometric spatial resolution can provide valuable information in several fields, most of the current literature using rare-earth based nanothermometers report ensemble-averaged data. Neglecting individual characteristics of each nanocrystal (NC) may lead to important inaccuracies in the temperature measurements. In this work, individual Yb3+/Er3+ codoped yttria NCs are characterized as nanothermometers when embedded in different environments (air, water and ethylene glycol) using the same 5 NCs in all measurements, applying the luminescence intensity ratio technique. The obtained results show that the nanothermometric behavior of each NC in water is equivalent to that in air, up to an overall brightness reduction related to a decrease in collected light. Also, it was observed that the thermometric parameters from each NC can be much more precisely determined than those from the “ensemble” equivalent to the set of 5 single NCs. The “ensemble” parameters have increased uncertainties mainly due to NC size-related variations, which we associate to differences in the surface/volume ratio. Besides the reduced parameter uncertainty, it was also noticed that the single-NC thermometric parameters are directly correlated to the NC brightness, with a dependence that is consistent with the expected variation in the surface/volume ratio. The relevance of surface effects also became evident when the NCs were embedded in ethylene glycol, for which a molecular vibrational mode can resonantly interact with the Er3+ ions electronic excited states used in the present experiments. The methods discussed herein are suitable for contactless on-site calibration of the NCs thermometric response. Therefore, this work can also be useful in the development of measurement and calibration protocols for several lanthanide-based nanothermometric systems.

: a) Transmission Electron Microscopy (TEM) image of multiple Y 2 O 3 :Yb 3+ /Er 3+ NCs b) Size distribution of the Yb 3+ /Er 3+ codoped NCs. The particles can be found in sizes ranging from ≈ 70 nm to ≈ 150 nm with average value of 120 ± 20 nm. c) Diffractogram of the codoped Y 2 O 3 :Yb 3+ /Er 3+ and pristine Y 2 O 3 confirming the body centered cubic phase structure of the NCs.

SI1. Morphological characterization of Y 2 O 3 NCs
The sample preparation protocol leading to sparse single NCs on a glass coverslip has been already described by Galvão et al., in reference. 1 Their work contains SEM and TEM images of single NCs before and after the spin coating technique used for preparing the samples (see subsection 2.1 and Figure 1c) of reference 1 and Figure 4a) of reference 2 for another sample -Nd 3+ doped ytrria NCs following the same sample preparation protocol).
Their results show that the spin coating technique herein used leads to a majority of single particle sites spread all over the glass coverslip surface.     Table 1.

SI3. Thermal resolution of individual nanocrystals (NCs)
The thermal resolution of a thermometer (δT ) is defined as the minimum temperature change that the system is able to measure confidently, 3 thus it is a function of the LIR (R).
One can expand δT in Taylor's Series and truncate to the first non-vanishing term, which results in According to Brites et. al., 3 the uncertainty in the determination of the LIR (δR) can be calculated either by measuring a set of LIR values in the same experimental condition, making a histogram and calculating the standard deviation; or by propagating the uncertainties from the signal-to-noise ratio of the detection system. Even though the above-mentioned procedures are well-established, they are not suitable to individual calibration in our system.
A different approach which can also be used to gain more physical intuition about the system is discussed below.
As reported in the main text, the calibration of the thermometer is made by fitting the LIR vs. Temperature data and obtaining two parameters: α, and β (equation 2 in the main manuscript). The fitting is performed by standard linear regression, in which it is also possible to calculate the variance-covariance matrix, defined by 4 is the variance of the variable γ = α or β, where E[γ] is its the expected value, and is the covariance of the two variables α and β.
If the two variables are independent random variables, the covariance between them must vanish, and the variance-covariance matrix becomes diagonal. Thus each element of the diagonal completely characterizes the statistical properties of its corresponding parameter.
In the opposite situation, where the variables are not independent, the Σ matrix is not diagonal, but still symmetric.
In the present work, however, by performing the linear regression via the Least Mean Square algorithm, it was observed that the covariance matrices for all thermometers fittings are not diagonal, which means that α and β are correlated. The physical meaning for this correlation relies on the choice of the Boltzmann Law to model the photophysical dynamics of the system. As discussed in a very recent work, 5 the authors used the same codoped system in a NaYF 4 matrix and showed that the dynamics of non-radiative absorption and decay of the Er 3+ ions leads to a deviation of the Intensity Ratio from the Boltzmann law, for the same thermally coupled levels used in this work.
Naturally, the β parameter depends on the radiative decay rates (equation 5 in the main manuscript), but the authors showed that the ∆E parameter evaluated by the Boltzmann Law is actually not the spectroscopic value, but an apparent value that depends also on the radiative decay rates. Therefore, α and β must be correlated in experiments. An important consequence is that the error propagation to obtain derived quantities as the thermal resolution must consider the off-diagonal terms in Σ.
In order to determine δR, it is possible to set that, for a fixed temperature T 0 , X 0 = 1/T 0 and R as R(α, β). Thus, the δR can be written in matrix form 4 The subsequent calculations of all related quantities presented in Table 1 follows from standard error propagation.

SI4. "Ensemble" averages and thermal resolution
The "ensemble" values presented in Table 1 for ∆E , β and S R were extracted from the mean of the values for the five selected NCs. The error bars for those values were defined as the standard deviation from the mean for each parameter.
The thermal resolution for the ensemble (δT (e) ) was obtained according to the references ( 1,5 ), being applicable for systems with multiple micro/nano-thermometers at a fixed temperature (T 0 ) and depends on the standard deviation (σ) of the LIR values for the studied NCs through where the superscript (e) holds for the ensemble mean values for R and S R . The values for δT (e) presented in Table 1 are higher than the obtained thermal resolution for the individual NCs because the dispersion of the LIR values for the set of nanothermomethers leads to a high standard deviation.